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Theorem mircgrextend 28525
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28328. (Contributed by Thierry Arnoux, 4-Oct-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Baseβ€˜πΊ)
mirval.d βˆ’ = (distβ€˜πΊ)
mirval.i 𝐼 = (Itvβ€˜πΊ)
mirval.l 𝐿 = (LineGβ€˜πΊ)
mirval.s 𝑆 = (pInvGβ€˜πΊ)
mirval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
mirtrcgr.e ∼ = (cgrGβ€˜πΊ)
mirtrcgr.m 𝑀 = (π‘†β€˜π΅)
mirtrcgr.n 𝑁 = (π‘†β€˜π‘Œ)
mirtrcgr.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
mirtrcgr.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
mirtrcgr.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
mirtrcgr.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
mircgrextend.1 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ))
Assertion
Ref Expression
mircgrextend (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΄)) = (𝑋 βˆ’ (π‘β€˜π‘‹)))

Proof of Theorem mircgrextend
StepHypRef Expression
1 mirval.p . 2 𝑃 = (Baseβ€˜πΊ)
2 mirval.d . 2 βˆ’ = (distβ€˜πΊ)
3 mirval.i . 2 𝐼 = (Itvβ€˜πΊ)
4 mirval.g . 2 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 mirtrcgr.a . 2 (πœ‘ β†’ 𝐴 ∈ 𝑃)
6 mirtrcgr.b . 2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
7 mirval.l . . 3 𝐿 = (LineGβ€˜πΊ)
8 mirval.s . . 3 𝑆 = (pInvGβ€˜πΊ)
9 mirtrcgr.m . . 3 𝑀 = (π‘†β€˜π΅)
101, 2, 3, 7, 8, 4, 6, 9, 5mircl 28504 . 2 (πœ‘ β†’ (π‘€β€˜π΄) ∈ 𝑃)
11 mirtrcgr.x . 2 (πœ‘ β†’ 𝑋 ∈ 𝑃)
12 mirtrcgr.y . 2 (πœ‘ β†’ π‘Œ ∈ 𝑃)
13 mirtrcgr.n . . 3 𝑁 = (π‘†β€˜π‘Œ)
141, 2, 3, 7, 8, 4, 12, 13, 11mircl 28504 . 2 (πœ‘ β†’ (π‘β€˜π‘‹) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 28501 . . 3 (πœ‘ β†’ 𝐡 ∈ ((π‘€β€˜π΄)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 28331 . 2 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼(π‘€β€˜π΄)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 28501 . . 3 (πœ‘ β†’ π‘Œ ∈ ((π‘β€˜π‘‹)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 28331 . 2 (πœ‘ β†’ π‘Œ ∈ (𝑋𝐼(π‘β€˜π‘‹)))
19 mircgrextend.1 . 2 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 28323 . . 3 (πœ‘ β†’ (𝐡 βˆ’ 𝐴) = (π‘Œ βˆ’ 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 28500 . . 3 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜π΄)) = (𝐡 βˆ’ 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 28500 . . 3 (πœ‘ β†’ (π‘Œ βˆ’ (π‘β€˜π‘‹)) = (π‘Œ βˆ’ 𝑋))
2320, 21, 223eqtr4d 2775 . 2 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜π΄)) = (π‘Œ βˆ’ (π‘β€˜π‘‹)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 28328 1 (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΄)) = (𝑋 βˆ’ (π‘β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6543  (class class class)co 7413  Basecbs 17174  distcds 17236  TarskiGcstrkg 28270  Itvcitv 28276  LineGclng 28277  cgrGccgrg 28353  pInvGcmir 28495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-trkgc 28291  df-trkgb 28292  df-trkgcb 28293  df-trkg 28296  df-mir 28496
This theorem is referenced by:  mirtrcgr  28526
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